An operator-theoretic approach to the mixed-sensitivity minimization problem

We consider the mixed-sensitivity minimization problem (scalar case). It gives rise to the so-called two-block problem on the algebra H^∞; we analyze this problem from an operator point of view, using Krein space theory. We obtain a necessary and sufficient condition for the uniqueness of the solution and a parameterization of all solutions in the non-uniqueness case. Moreover, an interpolation interpretation is given for the finite-dimensional case.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

An H-minimax approach to the design of robust control systems, part II: All solutions, all-pass form solutions and the ‘best’ solution

We characterize all solutions to a robustness optimization problem as the solutions of a two-parameter interpolation problem. From this characterization it is easy to show that an all-pass form solution always exists as long as a solution exists. We also study the possibility of using non-all-pass form solutions and by introducing other optimization objectives (motivated by improvements in disturbance rejection and robust stability) we search for the 'best' solution.     

A Hamiltonian-based solution to the two-block H problem for general plants in H and rational weights

A complete skew-Toeplitz-type solution to the two-block H^∞ problem for infinite-dimensional stable plants with rational weights is derived with a basis-free proof. The solution consists of one Riccati equation with a rank criterion for a transcendental function of a certain Hamiltonian. This gives a natural extension of the well-known formula for the one-block case. An example is given to illustrate the result.     

On the mixed sensitivity minimization for systems with infinitely many unstable modes

In this note we consider a class of linear time invariant systems with infinitely many unstable modes. By using the parameterization of all stabilizing controllers and a data transformation, we show that controllers for such systems can be computed using the techniques developed earlier for infinite dimensional plants with finitely many unstable modes.     

sensitivity minimization for delay systems

This paper announces results on the problem of feedback compensator design for ^∞ norm weighted sensitivity minimization when the plant contains a delay in the input. A complete solution is presented for the case of one pole/zero weighting function and a single-input/single-output plant for stable minimum-phase rational part. Generalizations and proofs will be published elsewhere.     

On decoupling the H-optimal sensitivity problem for products of plants

In this note we show how to solve the H^∞-optimal sensitivity problem for a SISO plant P(s) = P₁(s)P₂(s), given the solutions for P₁(s), P₂(s). This allows us to solve the problem for systems of the form e^−hsP₀(s), where P₀(s) is the transfer function of a stable, LTI, finite dimensional system.     

A Hamiltonian-based solution to the mixed sensitivity optimization problem for stable pseudorational plants

This paper considers the mixed sensitivity optimization problem for a class of infinite-dimensional stable plants. This problem is reducible to a two- or one-block H^∞ control problem with structured weighting functions. We first show that these weighting functions violate the genericity assumptions of existing Hamiltonian-based solutions such as the well-known Zhou–Khargonekar formula. Then, we derive a new closed form formula for the computation of the optimal performance level, when the underlying plant structure is specified by a pseudorational transfer function.     

L 1 Sensitivity minimization for plants with commensurate delays

In this paper we consider the problem ofL ¹ sensitivity minimization for linear plants with commensurate input delays. We describe a procedure for computing the minimum performance, and we characterize optimal solutions. The computations involve solving a one-parameter family of finite-dimensional linear programs. Explicit solutions are presented for important special cases.Notation X ^* Dual space of a normed linear spaceX - All elements inS with norm 1 - S The annihilator subspace defined as . - S The annihilator subspace defined as . - BV(X) Functions of bounded variation onX - C ₀(X) Continuous function on a locally compact spaceX such that for all > 0, {x X¦ ¦f(x)¦s is compact - C ^N (a, b) Vectors of continuous functions on (a, b) The authors acknowledge support from the Army Research Office, Center for Intelligent Control, under grant DAAL03-86-K-0171, and the National Science Foundation, under grant 8810178-ECS.     

Some remarks on Hamiltonians and the infinite-dimensional one block H problem

In this note, a simple proof is given for the Hamiltonian solution to the H^∞ optimal sensitivity for plants with arbitrary inner transfer function. The approach combines skew Toeplitz theory and state-space representations, and gives rise to a straightforward and basis-free methodology.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Closed formulae for a parametric-mixed-sensitivity problem for Pritchard-Salamon systems

The parametric-mixed-sensitivity problem is posed for the class of irrational transfer matrices with a realization as a Pritchard-Salamon system. An exact solution is obtained in terms of two uncoupled, standard Riccati equations.     

On pure H reduced-order dynamic controllers: an erratum

There are at least two approaches advocated to obtain a pure H^∞ reduced-order dynamic controller for a given augmented plant. One approach is to eliminate completely the H² aspect from a standard H²/H^∞ setting. A second approach is to equate the H² aspect with the H^∞ aspect in that same setting. This paper invalidates the first approach but affirms the second approach and produces the correct equations resulting therefrom.     

Output feedback decentralized control of large-scale systems using weighted sensitivity functions minimization

This paper considers the problem of achieving stability and desired dynamical transient behavior for linear large-scale systems, by decentralized control. It can be done by making the effects of the interconnections between the subsystems arbitrarily small. Sufficient conditions for stability and diagonal dominance of the closed-loop system are introduced. These conditions are in terms of decentralized subsystems and directly make a constructive H_∞ control design possible. A mixed H_∞ pole region placement is suggested, such that by assigning the closed-loop eigenvalues of the isolated subsystems appropriately, the eigenvalues of the overall closed-loop system are assigned in desirable range. The designs are illustrated by an example.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

An H-minimax approach to the design of robust control systems

A realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices. A solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone.     

Achievable H performance in sampled-data smoothing: beyond the -barrier

The lifting technique is a powerful tool for handling the periodically time-varying nature of sampled-data systems. Yet all known solutions of sampled-data H^∞ problems are limited to the case when the feedthrough part of the lifted system, , satisfies , where γ is the required H^∞ performance level. While this condition is always necessary in feedback control, it might be restrictive in signal processing applications, where some amount of delay or latency between measurement and estimation can be tolerated. In this paper, the sampled-data H^∞ fixed-lag smoothing problem with a smoothing lag of one sampling period is studied. The problem corresponds to the a-posteriori filtering problem in the lifted domain and is probably the simplest problem for which a smaller than performance level is achievable. The necessary and sufficient solvability conditions derived in the paper are compatible with those for the sampled-data filtering problem. This result extends the scope of applicability of the lifting technique and paves the way to the application of sampled-data methods in digital signal processing.     

The use of representations of H in placing classical control problems in a modern control framework

Motivated by Zames' work on optimal sensitivity, it is shown that there exists in functional analysis a foundation, based upon the theory of representations of H^∞, upon which results may be developed to enable a range of classical control problems to be placed in a modern control framework.     

On inner-outer factorization

The calculation of an inner-outer factorization of a proper, rational matrix by state-space methods is considered. Particular attention is given to the extraction of the inner factor, so that a minimal realization is obtained for this factor.     

Optimum performance levels for minimax filters, predictors and smoothers

For both discrete and continuous-time linear time-varying systems, we obtain the achievable performance levels for minimax filters, predictors and smoothers, in terms of the finite escape times of some related (discrete and continuous-time) Riccati equations. Our game-theoretic approach also yields an alternative derivation for the corresponding minimax estimators which were first obtained in [1]. They are all Bayes estimators with respect to particular Gaussian distributions, and admit recursive structures.